This is called a hard fork, but there is little to worry about for the end user. It is a bit similar to updating your iPhone. Over time, most iPhone users do it, and during the transition, both old and new versions still get software updates and can make phone calls and text with each other. However, only users with the latest version can use new features like Apple Pay.
It is still possible that these two versions of Bitcoin - Classic and Core - reunite together, and that a hard fork is not needed. Otherwise, we may see a small Bitcoin Core network operating mostly as an academic curiosity, and a large network running Bitcoin Classic, which could adopt any useful improvements developed in Bitcoin Core, as both are open source software.
Meanwhile, last Monday evening I paid my share of a dinner with bitcoin and we did not notice any delay!
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